Proving sin 2x = 2 sinx cosx with Maxima

Hi all,

Inspired by the book "A=B" by Zeilberger et al. I decided to try
the second method described at page 11 to prove a simple trignometric
equality.

The equality I tried is

    sin 2x = 2 sinx cosx .

The first step is to rewrite cos in terms of sin.
That is, I want to prove
                               2
    sin 2x = 2 sinx sqrt( 1-sin (x) )

in stead.

The next step is to substitute sin(x) with z.
The left hand side becomes

   f(z) :=  sin ( 2 asin(z) )  ,

and the right hand side becomes

   g(z) :=  2 z sqrt( 1 - z^2 ) .

Now f and g is to be differentiated wrt z. We
turn to Maxima:

First the left hand side:

(%i1) f(z):= sin(2*asin(z));
(%o1) 			    f(z) := SIN(2 ASIN(z))

(%i2) f1(z) := diff(f(z),z);
(%o2) 			    f1(z) := DIFF(f(z), z)

(%i3) f1(z);
			       2 COS(2 ASIN(z))
(%o3) 			       ----------------
					   2
				 SQRT(1 - z )


Then the right hand side:

(%i7) g(z) := 2*z*sqrt(1-z**2);
						 2
(%o7) 			   g(z) := 2 z SQRT(1 - z )

(%i8) g1(z) := diff(g(z),z);
(%o8) 			    G1(z) := DIFF(g(z), z)

(%i9) g1(z);
						 2
				     2	      2 z
(%o9) 			 2 SQRT(1 - z ) - ------------
						    2
					  SQRT(1 - z )


The problem is now to prove that f1(z) is equal to g1(z).

The book uses in an example the Maple expression

    normal(simplify(expand(g1**2-f1**2))) .

I have tried various Maxima-functions, but I can't figure out
what to write in order to reduce f1-g1 (or f1^2-g2^2) to 0 in
my example.

Any suggestions?

--
Jens Axel Søgaard
.

Relevant Pages

  • Re: Proving sin 2x = 2 sinx cosx with Maxima
    ... Jens Axel Søgaard schrieb: ... the second method described at page 11 to prove a simple trignometric equality. ... The first step is to rewrite cos in terms of sin. ...
    (sci.math.symbolic)
  • Re: Proving sin 2x = 2 sinx cosx with Maxima
    ... the second method described at page 11 to prove a simple trignometric equality. ... Rewriting cosine in terms of sine, ... occurences of sine and cosine. ...
    (sci.math.symbolic)
  • Re: Computer method for generating sine waves
    ... formula yields a whole lot of 0.0s: ... What information is missing in the second method to get it to work? ... Might depend on the implementation of sin() in your library. ... the results of sinwill loose precision. ...
    (sci.math)
  • Computer method for generating sine waves
    ... What information is missing in the second method to get it to work? ... The poster of the epsilon method suggested that the sin() table lookup ... not taking into account an accurate time methodology, ...
    (sci.math)
  • Re: Computer method for generating sine waves
    ... What information is missing in the second method to get it to work? ... The poster of the epsilon method suggested that the sin() table lookup ... not taking into account an accurate time methodology, ...
    (sci.math)